Find the sum of the first $35$ terms in this geometric series: $8 -12 + 18...$ Choose 1 answer: Choose 1 answer: (Choice A) A $-8.69\cdot10^{30}$ (Choice B) B $-3{,}106{,}363.96$ (Choice C) C $ 4{,}659{,}553.94 $ (Choice D) D $ 7{,}765{,}917.90 $
Solution: Getting started We're dealing with a geometric series because each term is multiplied by $-1.5$ to get the next term. We need a formula to compute the sum of the terms. Formula for geometric series The sum $S_n$ of a finite geometric series is $S_n = \dfrac{a_1(1-r^n)}{1-r}$ where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {8})$ and the number of terms $(n = {35})$ are given in the question. The common ratio $r$ is ${-1.5}$ because each term is multiplied by ${-1.5}$ to get the next term. [How did we find the common ratio r?] Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac{a_1(1-r^n)}{1-r} \\\\ S_{{35}}&=\dfrac{{8}(1-\left({-1.5}\right)^{{35}})}{1-\left({-1.5}\right)} \\\\ S_{{{35}}} &\approx 4{,}659{,}553.94 \end{aligned}$ The answer $ 4{,}659{,}553.94 $